Volume of Tubes and Concentration of Measure in Riemannian Geometry

TL;DR

This paper explores the relationship between volume of tubes around submanifolds and measure concentration phenomena in Riemannian manifolds, providing formulas and characterizations for specific cases.

Contribution

It introduces a general volume formula for tubes around submanifolds and analyzes concentration phenomena, especially in symmetric spaces and codimension one cases.

Findings

Explicit concentration in codimension one cases

General volume formula for tubes around submanifolds

Characterizations of concentration loci using Wasserstein and Box distances

Abstract

We investigate the notion of concentration locus introduced in \cite{CacUrs22}, in the case of Riemann manifolds sequences and its relationship with the volume of tubes. After providing a general formula for the volume of a tube around a Riemannian submanifold of a Riemannian manifold, we specialize it to the case of totally geodesic submanifolds of compact symmetric spaces. In the case of codimension one, we prove explicitly concentration. Then, we investigate for possible characterizations of concentration loci in terms of Wasserstein and Box distances.